nous 726 lr - department of...

NOUS 43:4 (2009) 742–775

Dimensional Explanations

MARC LANGE

The University of North Carolina at Chapel Hill

1. Introduction

Consider a small stone tied to a string and whirled around at a constant speed(figure 1). Why, one might ask, is the tension in the string proportional tothe square of the stone’s angular speed (rather than, say, varying linearly withits speed, or varying with the cube of its speed)? This is a why-question—arequest for a scientific explanation.

Here, perhaps, is an answer to the why-question. To a sufficiently goodapproximation (for example, when the string’s mass and the stone’s sizeare negligible), the tension F depends on no more than the following threequantities: (i) the stone’s mass m, (ii) its angular speed ω, and (iii) the string’slength r. These three quantities suffice to physically characterize the system.In terms of the dimensions of length (L), mass (M), and time (T), F ’s di-mensions are LMT−2, m’s dimension is M, ω’s dimension is T−1, and r’sdimension is L. In tabular form,

F m ω r

L 1 0 0 1

M 1 1 0 0

T −2 0 −1 0

If (to a sufficiently good approximation) F is proportional to mα ωβ rγ ,then the table gives us three simple simultaneous equations (one from eachline) with three unknowns:

C© 2009 Wiley Periodicals, Inc.

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Dimensional Explanations 743

m

F

ωω

r

Figure 1

From L: 1 = γ

From M: 1 = α

From T: −2 = −β

Since α = 1, β = 2, and γ = 1, F is proportional to mω2r.Apparently, then, dimensional considerations suffice to explain the ex-

planandum: F must be proportional to ω2, rather than to ω or to ω3, becauseF is a function of m, ω, and r alone, and among these three quantities, ω is theonly one capable of supplying F ’s time dimension (T−2). If such an argumentamounts to a “dimensional explanation”, then this variety of explanation hasbeen unjustly neglected in the enormous philosophical literature on scientificexplanation.

Of course, dimensional analysis is well-known in physics and engineeringas a shortcut (Birkhoff 1950, Bridgman 1931, Langhaar 1951, Sedov 1959).If you have identified all of the relevant quantities characterizing a givenphysical system, including the dimensional constants (such as the speed oflight c and Newton’s gravitational-force constant G), and if you know theirdimensions, then by dimensional considerations alone, you may learn a con-siderable amount about the relation holding among those quantities—as wejust did regarding the string tension’s relation to m, ω, and r. A physics stu-dent who has forgotten whether the tension is proportional to ω, ω2, or ω3

could figure it out on purely dimensional grounds (as long as she rememberswhich quantities are relevant to F and their dimensions).

However, I have rarely seen dimensional arguments characterized as pos-sessing explanatory power.1 Yet they appear to do so. These putative dimen-sional explanations form the subject of this paper.

Consider a derivative law2 (such as the example we just saw: that the cen-tripetal force on a small body of mass m moving uniformly with angular speedω in a circular orbit of radius r is proportional to mω2r). This law followsfrom various, more fundamental laws. However, I shall argue, a derivativelaw’s dimensional explanation can supply a kind of understanding that isnot provided by its derivation from more fundamental laws. For instance, itsdimensional explanation may reveal which features of the more fundamental

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laws entailing it are in fact responsible for it and which features are explana-torily irrelevant to it. Likewise, different features of the same derivative lawmay receive quite different dimensional explanations, whereas they are nottraced to different sources in the law’s derivation from more fundamentallaws. A dimensional explanation may reveal some features of a given deriva-tive law to be independent of some features of the more fundamental lawsentailing it.

A derivative law concerning one physical system may turn out to be similarin form to a derivative law concerning a physically unrelated system. A sep-arate derivation of each derivative law from various, more fundamental lawsmay explain each. However, neither one of the derivative laws explains theother, and the separate derivations may fail to identify any feature commonto the two systems as responsible for the similarity of the two derivative lawsconcerning them. The pair of explanations may thus portray the similaritybetween the laws as a kind of “coincidence” (albeit a naturally necessaryone). On the other hand, the two systems may be dimensionally similar incertain respects. The dimensional features common to the two systems mayaccount for the similarity in the derivative laws concerning them. In thatevent, a dimensional explanation succeeds in unifying what separate deriva-tions from more fundamental laws fail to unify. The dimensional explanationthen correctly characterizes the derivative laws’ similarity as no coincidence,but rather as explained by the dimensional architecture common to the twosystems.3

Likewise, a dimensional explanation may point to certain dimensionaldifferences between two physical systems as responsible for certain differencesbetween the derivative laws concerning each of them. No such explanationsof those differences are supplied by separate derivations of the two derivativelaws from various, more fundamental laws.

Dimensional thinking not only yields new explanations of antecedentlyappreciated phenomena, but also (I shall argue) identifies new phenomenato explain—phenomena that can be expressed only in dimensional terms.Furthermore, having argued that dimensional explanations can reveal a givenderivative law to be independent of certain features of the more fundamentallaws entailing it, I will argue that in some dimensional explanations, partof what explains a derivative law is precisely its independence from certainfeatures of those more fundamental laws. Finally, I will suggest that somedimensional explanations may proceed from “meta-laws” that impose variousconstraints on first-order laws ( just as symmetry principles in physics areusually taken as constraining force laws and other first-order laws).

Dimensional explanation may not only constitute an important and over-looked variety of scientific explanation, but also help us to understand howa derivative law’s scientific explanation differs from its mere deduction frommore fundamental laws.


2. A Dimensional Explanation of a Derivative Law May RevealCertain Features of the More Fundamental Laws Entailing

It to be Explanatorily Irrelevant to It

Consider a planet of mass m orbiting a star of mass M, feeling (to a suf-ficiently good approximation) only the star’s gravitational influence and or-biting with period T in a circular orbit of radius r. Let’s focus solely on T ’srelation to r: that T ∝ r3/2. (The symbol “∝” means “is proportional to”.)We shall now compare two possible explanations of this proportionality.

Our first candidate explanation derives the explanandum from more fun-damental laws: Newton’s laws of motion and gravity.4 These laws tell usthat that F = ma and F = GMm/r2, where F is the force on the planetand a is the planet’s acceleration. A body undergoing circular motion at aconstant speed v experiences an acceleration a of v2/r towards the center.Hence,

GMm/r 2 = mv2/r,

and so

GM/r 2 = v2/r,

and therefore

GM/r = v2.

For a circular orbit with circumference c = 2πr, the period equals the distancec covered in one revolution divided by the speed v. That is,

T = 2πr/v,

and so

T2 = 4π2r 2/v2.

By inserting the expression for v2 derived from Newton’s laws, we find

T2 = 4π2r 2/(GM/r ) = 4π2r 3/GM.

Perhaps we have thereby explained why T ∝ r3/2.Now I shall work towards giving a “dimensional explanation” of the fact

that T ∝ r3/2. The explanans will be that T stands in a “dimensionallyhomogeneous” relation to some subset of m, M, G, and r. A relation is

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“dimensionally homogeneous” if and only if the relation holds in any systemof units for the various fundamental dimensions of the quantities so related.5

Of course, r’s numerical value in meters will differ from its numerical valuein feet. But the same relation can hold among the quantities’ numericalvalues no matter what units are used to measure those quantities, even if thenumerical values so related differ in different units. I’ll say a bit more aboutdimensional homogeneity (and the explanans in a dimensional explanation)in sections 6 and 7.

The concept of dimensional homogeneity presupposes the concept of twounits (e.g., grams and slugs) counting as different ways of specifying thesame quantity (e.g., an object’s mass). Two units so qualify only if necessar-ily whenever x is the numerical value in one unit and y is the correspondingnumerical value in the other unit, y = cx for some constant c > 0—the“conversion factor” between the units (Bridgman 1931: 18–21). This con-dition is motivated by the idea that two units are not means of specifyingthe same quantity if, by changing between the units, the ratio between twomeasurements is not preserved.6 For example, since grams and slugs are unitsfor expressing the same quantity (mass), my measure in grams is twice myson’s measure in grams if and only if my measure in slugs is twice my son’smeasure in slugs.7

Suppose v = f(s, t, u, . . .) is continuous and dimensionally homogeneous,where s, t, u, v, . . . are the (positive-valued) quantities expressed in one systemof units. Suppose we now use a new system of units for these quantities,involving positive-valued conversion factors c, d, e, . . . , respectively; s whenconverted into the new units becomes cs, t becomes dt, u becomes eu, andso forth. Then since the equation is dimensionally homogeneous, f(cs, dt,eu, . . .) must equal whatever v becomes when converted into the new systemof units, which must be v multiplied by some function φ of c, d, e, . . . (whichgives the conversion factor for the quantity expressed by v). That is, f(cs, dt,eu, . . .) = φ(c, d, e . . .) f(s, t, u, . . .). It can be shown (see, e.g., Bridgman 1931:21–2, Birkhoff 1950: 87–8, Luce 1959: 87, Ellis 1966: 204) that this conditionholds for any new system of units if and only if f(s, t, u, . . .) is proportionalto sα tβ uγ . . . , where the constant of proportionality and the exponents aredimensionless constants—as long as there is no dimensionless combinationof s, t, u, . . . . (If there is, then f(s, t, u, . . .) is proportional to sα tβ uγ . . . timessome function of the dimensionless combination(s). Dimensional analysisalone is insufficient to identify the function.)

Let us return to the dimensional explanation of the fact that T ∝ r3/2. Theexplanans was that T stands in some “dimensionally homogeneous” relationto (some subset of ) m, M, G, and r. As we have just seen, this entails that Tis proportional to mα Mβ Gγ rδ (times some function of M/m). If distance(L), mass (M), and time (T) are taken as the fundamental dimensions, thenour table is as follows:


T m M G r

L 0 0 0 3 1

M 0 1 1 −1 0

T 1 0 0 −2 0

The table gives us three simultaneous equations (one from each line) withfour unknowns:

From L: 0 = 3γ + δ

From M: 0 = α + β − γ

From T: 1 = −2γ

From the L and T equations, it follows that γ = −1/2 and δ = 3/2. So T isproportional to r3/2, which is what we were trying to explain.

How does this “explanation” relate to the derivation (given earlier) ofthe T ∝ r3/2 law from more fundamental laws? One possibility is that thedimensional argument is not explanatory: although T ∝ r3/2 is entailed bythe fact that T stands in a dimensionally homogeneous relation to (somesubset of ) m, M, G, and r, both of these facts are explained by F = ma andF = GMm/r2. The supposed explanans and explanandum of a “dimensionalexplanation” have a common origin; one is not responsible for the other.

But dimensional arguments appear explanatory. Why might one insist thatnevertheless, they are not? Perhaps because a dimensional argument cannotyield more than various proportionalities—for instance, that T ∝ r3/2 andT ∝ G −1/2, whereas more fundamental laws entail the complete equation(T = 2π

√(r3/GM)), including the values of dimensionless constants of pro-

portionality (such as the 2π in this case) and that T is independent of theplanet’s mass (m).8 This difference might seem to suggest that whereas thederivation from more fundamental laws is explanatory, the dimensional argu-ment is not. However, I see no reason why the dimensional argument wouldhave to explain the entire equation above in order to explain why T ∝ r3/2 andT ∝ G −1/2.

I shall suggest that the dimensional argument is explanatory. Indeed, itreveals that as far as explaining T ∝ r3/2 is concerned, certain features of themore fundamental laws are idle. For example, T ∝ r3/2 depends neither onG’s specific value nor on those features of F = ma and F = GMm/r2 capturedby the M equation. Rather, T ∝ r3/2 arises from G’s being the only source ofT dimensions from among m, M, G, and r, and r’s being the only remainingsource of L dimensions to compensate for the L dimensions from G. The M

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equation does not figure in the argument. The explanans is then merely thatT stands in a relation to (some subset of ) m, M, G, and r—a relation thatis dimensionally homogeneous in terms of L and T. This explanans capturesthe only feature of the fundamental laws that is explanatorily relevant toT ∝ r3/2.

On this view, T ∝ r3/2’s derivation from F = ma and F = GMm/r2 con-tains elements that are otiose as far as explaining T ∝ r3/2 is concerned.The derivation is therefore like the famous counterexample from Kyburg(1965) to Hempel and Oppenheim’s D-N model: we cannot explain why agiven sample dissolved by saying that it was table salt, hexed (i.e., a personwearing a funny hat waved a wand and mumbled something over it), andplaced in water, and that as a matter of natural necessity, all hexed samplesof table salt dissolve when placed in water.9 To illustrate the independenceof T ∝ r3/2 from the features of F = ma and F = GMm/r2 expressed by theM equation, let us suppose that (contrary to natural law) the gravitationalforce had been proportional to M2m2 rather than to Mm. Then although G’sM dimension would have been different, its L and T dimensions would not.Thus, the L and T lines in the table would have been no different, and soT ∝ r3/2 would still have held.

The derivation from F = ma and F = GMm/r2 cannot reflect this inde-pendence, since there is no point in the course of that derivation after whichthe M dimensions of various quantities remain isolated from all of theirother dimensions. The derivation we saw earlier begins with the actual, morefundamental laws, not some generalizations thereof, and it simply churns on,deducing consequences of those laws until it generates the explanandum. Incontrast, the dimensional explanation begins by effectively teasing apart var-ious features of those more fundamental laws, and then it keeps these strandsapart, allowing the significance of each of them to be traced separately. Thusit can depict certain of these features as explanatorily relevant and others asmaking no contribution to the explanation.10

On this view, the dimensional argument explains why T ∝ r3/2 and thederivation from F = ma and F = GMm/r2 does not; it includes some factsthat are explanatorily irrelevant to T ∝ r3/2. Another way to put this pointinvolves turning to the explanans in the dimensional explanation and askingwhy it holds. Suppose the reason that T stands in a relation to (some subsetof ) m, M, G, and r that is dimensionally homogeneous in terms of L and Tis because of laws such as F = ma and F = GMm/r2 in all of their detail.In that case, all of those details would be explanatorily relevant to T ∝ r3/2.But as we have just seen, certain features of the fundamental laws are notreflected in the dimensional explanans and so do not help to explain whyT ∝ r3/2. The dimensional explanans is not explained by the fundamentallaws. Neither is explanatorily prior to the other. Rather, the dimensionalexplanans expresses exactly the fact about the fundamental laws that is re-sponsible for T ∝ r3/2.


In subsequent sections, I shall offer additional reasons to regard a di-mensional argument as explaining why some derivative law holds and assupplying understanding that cannot be supplied even in principle by thatderivative law’s deduction from more fundamental laws.

3. Different Features of a Derivative Law May Receive DifferentDimensional Explanations

I have just argued that a dimensional explanation may reveal certain aspectsof the more fundamental laws entailing a given derivative law to be explana-torily irrelevant to that law. That is because a dimensional explanation dis-regards certain aspects of the more fundamental laws (by considering onlytheir dimensional architecture) and treats certain other aspects separatelyfrom one another (by tracing individually each of several dimensions). Theway in which dimensional explanations tease apart the various dimensionshas another consequence: Different aspects of the same derivative law mayreceive different dimensional explanations. This is another respect in whicha law’s dimensional explanation may differ from the law’s deduction frommore fundamental laws. Just as a dimensional explanation may distinguishthe contributions (if any) made by different aspects of the more fundamentallaws, so also it may give separate explanations to different aspects of the lawbeing explained.

Here is an example. Consider a body with mass m lying on a smooth tableand subject to the force of a spring (figure 2).

Suppose we grasp the body and pull, stretching the spring to a distancexm beyond its equilibrium length, and then let go. The body then moves backand forth. Here are two why-questions concerning this case:

(1) Why is the body’s period of oscillation T proportional to√

(m/k), whereeach spring has a characteristic constant k?

(2) Why is T not a function of xm?

These two questions receive the same answer in terms of a deduction frommore fundamental laws: Newton’s second law of motion (F = ma) andHooke’s law (that the force F exerted by the spring on the body is towards

Figure 2

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the spring’s equilibrium position and, to a sufficiently good approximation, isproportional to the body’s displacement x from that position). The answer tothe two why-questions is that from these laws, we can deduce x(t), the body’sdisplacement as a function of time, which turns out to be an oscillation witha period T = 2π

√(m/k), so T is not a function of xm and is proportional

to√

(m/k). The derivation is as follows:

ma − F = 0

Let k be the constant of proportionality in Hooke’s law,

F = −kx.

So

ma + kx = 0,

and so

a + (k/m)x = 0.

This differential equation (a is x’s second-derivative with respect to time) issolved by

x(t) = xm cos(√

(k/m) t + φ),

which has a period of

2π√

(m/k).

Thus, as far as this deduction can reveal, the reason why T is not a functionof xm and is proportional to

√(m/k) is that—well, it just works out that way,

considering Newton’s second law and Hooke’s law.Let’s compare this deduction to a dimensional explanation. The explanans

is that (to a sufficiently good approximation) xm, k, and m suffice to com-pletely physically characterize the situation; they suffice (and may even bemore than enough) to form a quantity standing in a dimensionally homoge-neous relation to T . So here is our table:

T xm k m

L 0 1 0 0

M 0 0 1 1

T 1 0 −2 0


If T is proportional to xmα kβ mγ , then it follows from the table’s first line

that α = 0. In other words, the reason why T is not a function of xm isthat xm is the only quantity characterizing the system that involves an Ldimension, so there is nothing available to compensate for its L dimensionin order to leave us with T (which has no L dimension).

This dimensional explanation of T ’s independence from xm is distinctfrom the dimensional explanation of T ’s varying with

√(m/k). The latter

explanation is given by the M and T equations:

From M: 0 = β + γ

From T: 1 = −2β

Solving, we find that β = −1/2 and γ = 1/2, and so that T is proportional tok−1/2 m

1/2 , i.e.,√

(m/k).Different aspects of the derivative law thus receive different dimensional

explanations, since those explanations pull the various dimensions apart. Butthose different aspects of the derivative law all arise together at the conclusionof the law’s derivation from more fundamental laws.11 The first dimensionalexplanation identifies the specific feature of the fundamental laws that isresponsible for T ’s independence from xm, and the second dimensional ex-planation distinguishes this feature from the one that is responsible for T ’svarying with

√(m/k). By contrast, these features are not distinguished in the

derivation of x(t) = xm cos (√

(k/m) t + φ).

4. Dimensional Explanations Can Unify what Derivationsfrom More Fundamental Laws Cannot

In section 2, I argued that a dimensional explanation may reveal certainaspects of the more fundamental laws from which a given derivative lawfollows to be explanatorily irrelevant to that law. Hence, there should bepairs of laws that are entailed differently by more fundamental laws, butwhere those differences are confined to aspects of the more fundamentallaws that turn out to be explanatorily irrelevant according to the dimensionalexplanations of the two derivative laws. In that event, the two derivative lawsreceive the same dimensional explanation. In other words, we should expectdimensional explanations to unify derivative laws that are derived separatelyfrom more fundamental laws.

For example, consider a wave of pressure propagating in a fluid (or elasticsolid) surrounded by rigid walls (such as water in a rigid pipe). A compressionmoves through the fluid, so that at a given moment, there are alternatingregions of compression and rarefaction. Over time, a small region of the fluidalternately undergoes compression and rarefaction, as the elements of thefluid in that region are pushed together or spread apart. Each element of the

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fluid oscillates back and forth along the same line as the wave propagates.That is, a pressure wave (such as a sound wave) is “longitudinal”. Let uscompare this wave to the wave propagating down a stretched string, such asa guitar string held taut by tuning pegs and plucked at one end. That waveis “transverse” in that the elements of the string oscillate back and forth in adirection perpendicular to the direction of the wave’s propagation. (Assumethat the amplitude of the wave is small compared to the length of the string.)Despite the differences between these two cases, the wave’s speed v in thetwo cases is given by remarkably similar equations:

longitudinal: transverse:

v = √(B/ρ) v = √

(F/μ)

where ρ is the medium’s density, where μ is the string’s linear density,

B is its bulk modulus12 F is its tension

The analogy between ρ and μ is evident; the only difference between themreflects the fact that the medium in the longitudinal case occupies a volume,whereas the medium in the transverse case is a string (one-dimensional). Theanalogy between B and F may be less evident. A medium’s bulk modulusB reflects how much additional pressure must be exerted on the medium toreduce its volume by a certain fraction; if, by changing the pressure it feelsby dP, one changes its volume by dV in total, or by (dV/V ) per each unitof its volume V , then its bulk modulus is given by

B = −dP/(dV/V).

(The minus sign makes B a positive quantity, since if dP > 0, then dV < 0,considering that increased pressure brings decreased volume.) If the fluid’sbulk modulus is larger, then greater additional pressure is needed to compressthe fluid by a certain fraction. In this respect, the bulk modulus is like thestring’s tension; if the string is tighter, then greater force is needed to pluckit—that is, to make it bend (which requires lengthening it). In short,

√(B/ρ)

and√

(F/μ) are alike in that each takes the form√

(medium’s elasticity/density).

Why is there such a similarity between these physically dissimilar kindsof waves? A derivation of the transverse wave’s v from more fundamentallaws explains why that quantity equals

√(F/μ). An unrelated derivation of

the longitudinal wave’s v from more fundamental laws explains why thatquantity equals

√(B/ρ). Do these separate explanations for each wave’s v

explain the similarity in the two v equations—that is, do they together explainwhy the two waves’ velocities are each proportional to

√(medium’s elasticity/

density)? Only if this similarity in the waves’ v is in fact coincidental. Butit turns out not to be coincidental. Rather, as the dimensional explanation


z (the direction of propagation)

dz A

Figure 3

reveals, their similarity actually arises from a feature common to the twokinds of waves.

We can see this by starting with the longitudinal wave (figure 3). Take afluid element thin enough in the direction of the wave’s propagation that ithas uniform density and internal pressure (though its density and pressurechange as it becomes part of a compression or rarefaction). Let ρ be thefluid’s density when not disturbed by a wave passing through, and let theelement’s original width be dz. Each fluid element has an initial location zalong the length of the pipe, and an element’s displacement s(z,t) at time tfrom its original location z varies periodically in z and in t. That is the wave.The pipe has cross-sectional area A. The fluid element’s front and back wallswill be shifted from their equilibrium positions as a wave passes throughit. The change in the element’s length will be the change in its front wall’sposition minus the change in the rear wall’s position: s(z + dz) − s(z). Thechange dV in its volume will therefore be A [s(z + dz) − s(z)], and since B =−dP/(dV/V ), we have

dP = −B dV/V = −BA[s(z + dz) − s(z)]/A dz = −B ∂s/∂z,

and so

∂ P/∂z = −B ∂2s/∂z2.

The force pressing inward on the element’s front wall is A P(z + dz) in the−z direction, and the force pressing inward on the element’s rear wall isA P(z) in the +z direction, so the net force F on the element is A [P(z) –P(z + dz)], and so dF/dz = −A ∂P/∂z. Hence,

dF/dz = AB ∂2s/∂z2.

By Newton’s second law, the force dF on an element is given by its accelera-tion ∂2s/∂t2 times its mass dm = ρA dz, and so

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Figure 4

dF/dz = ρA∂2s/∂t2.

Equating our two expressions for dF/dz, we find

ρA∂2s/∂t2 = AB ∂2s/∂z2,

and so

∂2s/∂t2 = (B/ρ) ∂2s/∂z2.

This is a form of the wave equation, relating the way s(z,t) changes over timeat a given place to the way s(z,t) changes over place at a given time. It issolved by a wave propagating in the +z direction at speed v = √

(B/ρ). Sothat is why a longitudinal wave’s speed equals

√(B/ρ).

To explain the transverse wave’s speed, take a small segment AB of theplucked string that makes an angle θ (figure 4). Approximate AB as the arc ofa circle of radius R, so that AB’s length is Rθ and mass is μRθ . Since anglesα, β, and γ are congruent, and α is half of θ , the vertical component of thetension F at each end of the arc is Fsin(θ/2), so the two ends’s contributionssum to 2 Fsin(θ/2), which can be approximated as 2F(θ/2) = Fθ since θ issmall. From the wave’s viewpoint, the string is moving at speed v, and sincethe centripetal force for segment AB moving in a circle of radius R at speedv is mv2/R = μRθ v2/R = μθv2, we have

Fθ = μθv2,

and so

v = √(F/μ).

Thus we have explained why a transverse wave’s v equals√

(F/μ).


Each wave’s v is proportional to its√

(medium’s elasticity/density). Ifthis similarity in the two v equations were explained by the conjunction ofthese two derivations, then that similarity would amount to a coincidence.However, it is not, since a dimensional explanation reveals that the similarityarises from a common feature of the two waves. In so doing, the dimen-sional explanation unifies v’s proportionality to

√(B/ρ) for the longitudinal

wave with v’s proportionality to√

(F/μ) for the transverse wave. Take theexplanans in the dimensional explanation to be that (to a sufficiently goodapproximation) for each wave, density, elasticity, and wavelength suffice (andmay even be more than enough) to form a quantity standing in a dimension-ally homogeneous relation to v. In other words, B, ρ, and the wavelength λ

completely physically characterize the longitudinal-wave system, and F , μ,and λ completely physically characterize the transverse-wave system. Hence,v is proportional to ρα Bβ λγ or to μα Fβ λγ . Then the tables are as follows:

v ρ B λ

L 1 −3 −1 1

M 0 1 1 0

T −1 0 −2 0

v μ F λ

L 1 −1 1 1

M 0 1 1 0

T −1 0 −2 0

The M and T lines are identical in the two tables, and they give us twoequations in two variables (0 = α + β, −1 = −2β), from which it followsthat v is proportional to

√(B/ρ) and

√(F/μ), respectively. Although the

L line in the table is different in the two cases, that difference makes nodifference to α and β.13

Thus, unlike the derivations from more fundamental laws, the dimensionalexplanation traces the similarity (that each wave’s v is proportional to its√

(medium’s elasticity/density)) to a feature common to the two systems.The explanandum is revealed to be independent of the physical differencesbetween longitudinal and transverse waves.

Of course, in “unifying” the two cases, the dimensional explanation doesnot derive the two v equations from the very same facts. The dimensionalexplanation does not reveal a common explainer, since the expression for v

in a given case is explained by the dimensional architecture of that particularcase. The fact that there is a dimensionally homogeneous relation betweenv, B, ρ, and λ regarding longitudinal waves is plainly distinct from thefact that there is a dimensionally homogeneous relation between v, F , μ,and λ regarding transverse waves. The unification forged by dimensionalexplanations is not like the unity created by a common cause.

Nor is it like the unity created by Newton’s theory of gravity in revealingthat the moon is just a falling body—or that the tides, lunar and planetary

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motions, and falling bodies are all governed by the same equations. Lon-gitudinal waves are not at bottom the same as transverse waves. Moreover,the unity forged by dimensional explanations is not like the unity createdby Darwin’s theory of natural selection (according to Kitcher 1993) in thatit does not primarily involve showing various facts to be derivable throughthe same argument schemata or showing various questions to be answer-able through the same problem-solving patterns. Although all dimensionalexplanations employ dimensional arguments, not all of the facts explaineddimensionally are thereby unified with one another—unlike all of the factsabout anatomy, physiology, biochemistry, biogeography, embryology, and soforth that are unified with one another in virtue of fitting into Darwinianselectionist histories. (Moreover, Darwin’s theory unified various particularanatomical and biogeographical facts rather than various laws.)

Rather, the dimensional explanation unifies the derivative laws concerningv for longitudinal and transverse waves by identifying features commonto both systems that account for the similarities between the two v laws.The dimensional explanation explains why the two v expressions are sosimilar: because of the dimensional architecture that the two systems share.To conjoin derivations of the two v laws from more fundamental laws maybe to explain the two v laws, but the conjunction fails to explain why theyare so similar since it inaccurately depicts their similarity as a coincidence.14

Instead, their similarity arises from common features of the fundamentallaws as they apply to the two systems—features captured by the explanansin the dimensional explanation.

A dimensional explanation, then, may identify the features shared by somephysically disparate cases that are responsible for their similarity in variousother respects. It may likewise identify the differences between physicallydisparate cases that are responsible for various other differences betweenthem. For example, consider a simple pendulum (a small, heavy body ofmass m suspended near Earth’s surface by a cord of invariable length land negligible mass and cross-section) in vacuo. The period T of its swing,when it has been released from a small angle, is (to a sufficiently goodapproximation) proportional to

√(l/g). Compare the simple pendulum to

the system depicted in figure 5: a spring of negligible mass hangs verticallynear Earth’s surface (where every freely falling body experiences the samedownward acceleration g from gravity), and we suddenly attach a body ofmass m to the spring. The spring stretches, and then the body moves up anddown for a little while, where this oscillation’s period T is proportional to√

(m/k). We might contrast these two and ask: Why is the pendulum’s T afunction of g whereas T in figure 5’s system is independent of g?

One way to try to answer this question is to derive the two T ’s from morefundamental laws. But this putative explanation fails to trace this particulardifference between the T ’s to some other particular difference between thetwo cases. Regarding figure 5’s system, we find ma + kx − mg = 0, and so the


Figure 5

governing differential equation there is a + (k/m) x − g = 0, which is simpleharmonic motion with period 2π

√(m/k). Regarding the simple pendulum,

we find ma + [mg/l] x = 0, and so the governing differential equation thereis a + (g/l) x = 0, which is simple harmonic motion with period 2π

√(l/g).

According to this putative explanation, the pendulum’s period depends ong, whereas the period of figure 5’s system does not, simply because differentforces are at work in the two cases. No more specific points of contrast areheld responsible.

However, in dimensional terms, the two cases are more readily contrasted.On the left is our table for the pendulum15 and on its right is our table forfigure 5’s system:

T g l m

L 0 1 1 0

M 0 0 0 1

T 1 −2 0 0

T g k m

L 0 1 0 0

M 0 0 1 1

T 1 −2 −2 0

The differences are confined to the third column. However, that is enoughto make a considerable difference. In figure 5’s system, T cannot depend ong because g is the only characteristic of the situation with an L dimension,so nothing is available to compensate for g’s L dimension to leave us withpure T (for the period). In contrast, the pendulum has its characteristic l tocompensate for g’s contribution to the L dimension. Here we have an answerto our why-question that separate derivations from more fundamental lawscannot supply.

Likewise, why does T depend on m in figure 5’s system, but not for apendulum? Because m is the only parameter characterizing the pendulum thathas an M dimension, whereas in figure 5’s system, k has an M dimension that

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can compensate for m’s contribution. Separate derivations of T in the twocases from more fundamental laws fail to point out that as long as m remainsthe only characteristic with an M dimension, m cannot take any responsibilityfor T . As we saw in connection with the T ∝ r3/2 law, derivations from theactual more fundamental laws fail to identify the specific features of theselaws that are responsible for the derivative law.

5. Dimensional Explanations Can Target New Explananda

Consider the respects of similarity and difference that (we have just seen)dimensional explanations do a nice job of explaining—for example, that v

is proportional to√

(medium’s elasticity/density) for both longitudinal andtransverse waves, or that T depends on m in figure 5’s system but not for apendulum. It is easy to recognize these respects of similarity and difference.To do so, we do not need to think about the systems in dimensional terms.However, along with giving us new answers to why-questions that presentthemselves independently of dimensional considerations, dimensional rea-soning also suggests new why-questions to ask. That is because dimensionalconsiderations pick out new respects of similarity and difference for us toask about.16

For example, consider a body at rest that begins to fall freely. In time t,it covers a distance s = 1/2 gt2. That is, t = √

(2s/g), so t is proportional to√(s/g). A freely falling body is physically not much like a pendulum. For

instance, the falling body’s motion is one-way, whereas the pendulum movesperiodically; t in the above equation is not the period of any repeated motion.Yet dimensionally, t is like the pendulum’s period T . Indeed, there is a dimen-sional similarity between the two systems: just as for a freely falling body, tis proportional to

√(s/g), so for a pendulum, T is proportional to

√(l/g).

Dimensional reasoning reveals this similarity to be no coincidence. Sup-pose that t for falling bodies stands in a dimensionally homogeneous relationto g, s, and m, and suppose also that T for pendulums stands in a dimen-sionally homogeneous relation to g, l, and m. Then the two systems have thesame dimensional architecture, so dimensionally similar relations must holdamong their parameters. In tabular form: the pendulum is on the left andthe freely falling body is on the right:

T g l m

L 0 1 1 0

M 0 0 0 1

T 1 −2 0 0

t g s m

L 0 1 1 0

M 0 0 0 1

T 1 −2 0 0


Because the two systems have the same dimensional architecture, the deriva-tive laws for the falling body’s t and the pendulum’s T must be dimensionallyanalogous.

A water wave having wavelength λ (traversing deep water, taking water tobe incompressible and nonviscous) is physically neither like a falling bodynor like a pendulum. Yet its period T is proportional to

√(λ/g).17 We could

ask why the derivative law for the water wave’s period is so similar to thelaws for the pendulum’s T and the falling body’s t, despite the physicaldifferences among these phenomena. Once again, this similarity among thethree derivative laws can be appreciated only in dimensional terms.

The dimensional explanation of the water wave’s period starts by takingthe gravitational acceleration (reflecting the restoring force on the waterdisplaced in the wave), the water’s density ρ, and the wavelength as sufficientto form a quantity standing in a dimensionally homogeneous relation to thewave’s period. In tabular form:

T g λ ρ

L 0 1 1 −3

M 0 0 0 1

T 1 −2 0 0

Although this table is not perfectly dimensionally identical to the previoustwo, its single difference from them (in the L dimension of the only character-istic with an M dimension) makes no difference: Because that characteristicis the only one with an M dimension, it cannot figure in the dimension-ally homogeneous relation. Since the three tables are otherwise identical, theresulting relations must be dimensionally analogous.

In view of the dimensional similarity among the three systems, it is in-evitable that the expressions for the freely falling body’s t, the pendulum’sT , and the water wave’s T are analogous. But that analogy can be appre-ciated only in dimensional terms; s, l, and λ are not otherwise analogous.That stands in contrast to the longitudinal and transverse wave example (insection 4), where the analogy between ρ and μ, and between B and F , couldbe appreciated without invoking dimensional considerations. (Indeed, theanalogs do not have the same dimensions.)

6. Dimensional Homogeneity

In a dimensional explanation, the explanans is that there exists a dimension-ally homogeneous relation between a given quantity and a subset of certainother quantities—a relation involving no other quantities besides (some of )those. A relation is “dimensionally homogeneous” if and only if the relation

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holds in any system of units for the fundamental dimensions of the quantitiesso related. Not all relations are dimensionally homogeneous. For example,that (right now) my son’s weight equals my age is not dimensionally homo-geneous; this relation holds only if my son’s weight is measured in poundsand my age is measured in years.

This example illustrates the falsehood of remarks like this, from a standardphysics textbook: “Any equation must be dimensionally consistent; that is, thedimensions on both sides must be the same.” (Resnick, Halliday, and Krane1992: 9). On the contrary, the numerical value of one quantity, measured inone unit, can be equal to the numerical value of another quantity, measuredin another unit. It might be suggested that “equation” in the above remarkrefers only to laws of nature, not mere accidents, and that (as Luce (1971:157) declares) any law of nature must be dimensionally consistent.18 But thatis not so; for example, if a body starting at rest at t = 0 moves by time t adistance s under uniform acceleration a, then (where v is its speed at t) v2 −2as = v − at. The two sides have different dimensions but are numericallyequal, as a matter of natural law, no matter what units are used throughoutfor distance and time.19 Both sides equal zero—as does F − ma, in yetother dimensions. Dimensional consistency is not necessary for dimensionalhomogeneity.20

Although not every relation is dimensionally homogeneous, every relationcan be transformed into a dimensionally homogeneous relation by addingfurther relata. For example, that my son’s weight in pounds equals my agein years entails that my son’s weight in any unit, times the reciprocal of thenumber of those units in 1 pound, equals my age in any unit, times thereciprocal of the number of those units in 1 year. This relation is dimen-sionally homogeneous, but it involves not only my son’s weight and my age,but also two “dimensional constants.” (They could be combined into one.)Familiar dimensional constants appearing in natural laws include the speedof light c and Newton’s gravitational-force constant G. The explanans in adimensional explanation specifies all of the quantities, including dimensionalconstants, that are eligible to participate in some relation. So the explanans(that there is a dimensionally homogeneous relation between quantity Q andno other quantities besides a subset of the following quantities . . .) is not atrivial truth. But it is a trivial fact that among the relations that obtain, someare dimensionally homogeneous.

The fact that any relation can be transformed into a dimensionally ho-mogeneous relation undermines any remark along these lines: “The fact thatpractically every law of physics is dimensionally invariant is certainly no ac-cident. We need an explanation of this invariance . . .” (Causey 1967: 30; forsimilar remarks, see Causey 1969: 256; Luce 1971: 151, and Krantz, Luce,Suppes, and Tversky 1971: 504–6). We need neither to appeal to some meta-physical analysis of natural lawhood nor to posit some meta-law of nature(e.g., that it is a law that all laws are dimensionally homogeneous)21 in order


to explain why all actual laws involve dimensionally homogeneous relations.Rather, we need note only that any law expressed in terms of a relation thatis not dimensionally homogeneous can also be expressed in terms of a re-lation that is dimensionally homogeneous.22 For instance, in middle schoolI was taught that it is a law of nature that the distance s in feet traversedin t seconds by a body falling freely from rest is equal to 16t2. This law isplainly not dimensionally homogeneous. But if we replace the pure number16 with the dimensional constant 16 feet per second2, then we will have adimensionally homogeneous relation.23

In any event, a particular dimensional explanation does not presupposesome general principle that all laws involve dimensionally homogeneous re-lations. Rather, the explanans in a given dimensional explanation is muchnarrower: that there obtains a relation between a given quantity and a subsetof several other quantities that is dimensionally homogeneous for a certainset of fundamental dimensions. This reference to certain dimensions is thenext topic I shall discuss.

7. Independence from Some More Fundamental LawsMay be Part of a Dimensional Explanans

In a dimensional explanation, the explanans is that certain quantities sufficeto produce a dimensionally homogeneous relation—which entails that cer-tain other quantities do not need to be added. Whereas a deduction of somederivative law from more fundamental laws may appeal to a law that specifiesthe value of a given dimensional constant, for example, a dimensional expla-nation of that derivative law may appeal to the fact that the explanandumwould still have held whatever that dimensional constant’s value.

Consider, for instance, a small sphere of radius r falling slowly througha fluid of viscosity η. It quickly reaches a steady speed, its “terminal veloc-ity,” which turns out to be proportional to gr2/η. This fact may be derivedhydrodynamically or dimensionally.

The hydrodynamic derivation begins with Newton’s second law of motion:the net force on the sphere equals its ma. When terminal velocity is reached,a = 0. The net force on the sphere is the sum of the downward gravitationalforce Fg, the upward buoyant force Fb, and the upward drag force Fd thatincreases with the sphere’s speed. By Archimedes’s principle, Fb equals theweight of a fluid volume equal to the body’s volume. Furthermore, Fd isgiven by Stokes’s law when the fluid flow around the body is not turbulent(roughly, for a small body at low speed). If the fluid’s density is ρ f and thesphere’s density is ρ s, then

Fg = mg = ρs(4/3)πr 3g

Fb = −ρf (4/3)πr 3g

Fd = −6πrvη.

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So

(4/3)πgr3(ρs − ρf ) = 6πrvη.

Hence

v = (2/9)gr2(ρs − ρf )/η.

Thus, we have derived that v is proportional to gr2/η.Now consider a dimensional explanation of this law. Suppose the ex-

planans is that ρ s, ρ f , r, g, and η suffice to characterize the system physically.Hence (since ρ f/ρ s is dimensionless, and so its contribution is inaccessiblefrom exclusively dimensional considerations), there is a dimensionally ho-mogeneous relation v = rα ρ s

β ηγ gδ f(ρ f/ρ s) for some function f.24 If thisrelation is dimensionally homogeneous in terms of three dimensions (suchas L, T, and M), then unhappily, a table yields three simultaneous equa-tions in four variables (α, β, γ , and δ). However, suppose (as Bridgman1931: 66 suggests) that this relation is dimensionally homogeneous in termsof four dimensions: L, T, M, and force (F). Notice that according to thisexplanans, the dimensional constant of proportionality k between force andmass x acceleration (F = kma) is not among the quantities standing in thedimensionally homogeneous relation. It bears no explanatory responsibilityfor the fact that v is proportional to gr2/η because whatever k’s value, thatvalue is the same for all three component forces, so the value does not affectthe conditions under which they are balanced, which is the condition withwhich we are concerned. No force is unbalanced, so no force is producingacceleration, so the rate of exchange between force and ma does not matter.25

Then we have four simultaneous equations in four variables. Our table is

v r ρs η g

L 1 1 −3 −2 0

M 0 0 1 0 −1

T −1 0 0 1 0

F 0 0 0 1 1

Our four equations are then

From L: 1 = α − 3β − 2γ

From M: 0 = β − δ


From T: −1 = γ

From F: 0 = γ + δ.

Hence: α = 2, β = 1, γ = −1, δ = 1, yielding v = (r2ρ sg/η) f(ρ f/ρ s).26 Thus,we have explained why v is proportional to gr2/η.

This equation is thus explained by the fact that there is a relation betweenv, ρ s, ρ f , r, g, and η that is dimensionally homogeneous using these fourdimensions and so holds for any value of k. Whereas F = ma launches the hy-drodynamic derivation, the dimensional explanation not only fails to employF = ma, but actually employs the fact that the dimensionally homogeneousrelation would still have held whatever the value of k where F = kma. In thedimensional explanation, part of the explanans is that the explanandum isindependent from a certain feature of a more fundamental law that is used todeduce it hydrodynamically. In other words (putting the point more provoca-tively), part of the explanans is a counterfactual (indeed, a counterlegal): thatthe explanandum would still have held, for any value of k.27

This aspect of some dimensional explanations is crucial to resolving avenerable dispute. In an early discussion of dimensional methods, Rayleigh(1915a) considered the rate at which heat passes from a hot wire to a coolerstream of air passing across it—or, idealizing and generalizing, the rate hof heat loss by a rigid body of infinite conductivity and presenting a lineardimension a to an ideal (i.e., incompressible, inviscid) fluid flowing at speedv around it, where the body’s temperature is kept constant and exceeds thefluid’s initial temperature (i.e., far upstream from the body) by θ . Let c bethe fluid’s specific heat per unit volume (i.e., the heat needed to raise thetemperature of a unit volume of fluid by one degree) and let κ be the fluid’sthermal conductivity (i.e., the rate of heat flow through a unit thicknessof the fluid per unit area and unit temperature difference). Rayleigh useddimensional analysis to arrive at an expression for h, though I suggest thatwe consider him as giving a dimensional explanation of that expression. Hisproposed explanans is that h stands in a dimensionally homogeneous relationto some subset of a, v, θ , c, and κ, where the fundamental dimensions areL, T, � (temperature), and Q (heat). If h is to have the same dimensions asaα vβ θγ cδ κε, and our table is

h a v θ c κ

L 0 1 1 0 −3 −1

T −1 0 −1 0 0 −1

� 0 0 0 1 −1 −1

Q 1 0 0 0 1 1

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then (Rayleigh argued) we have four simultaneous equations with fiveunknowns

From L: 0 = α + β − 3δ − ε

From T: −1 = −β − ε

From � : 0 = γ − δ − ε

From Q: 1 = δ + ε

yielding α = β + 1, γ = 1, δ = β, and ε = 1 – β. Therefore, h has thesame dimensions as aβ+1 vβ θ cβ κ1−β , i.e., as (avc/κ)β aθκ, and so h =aθκ f(avc/κ) for some unknown function f. We have thereby explained whythe rate of heat loss is proportional to the temperature difference, why it isthe same in cases involving different values of v and c but where vc is thesame, and so forth.

However, in an economical three-sentence reply, Riabouchinsky (1915)noted that temperature and heat have the same dimensions as energy, sothat we have here only three fundamental dimensions, not four. (The averagekinetic energy of random motion per molecule is related to temperature byBoltzmann’s constant, equal to 1.38 × 10−16 ergs/degree Kelvin, and the rateof exchange between heat and energy units is given by the mechanical equiv-alent of heat, equal to 4.184 Joules/calorie.) We thus have one fewer equa-tion, but the same number of unknowns, so dimensional analysis yields lessinformation—merely that h = aθκ F(v/κa2, ca3), for some unknown functionF. (This dimensional argument fails to explain why heat loss is the same incases where vc is the same, for instance.) Rayleigh (1915b) replied uncertainly:

The question raised by Dr. Riabouchinsky belongs rather to the logic than tothe use of the principle of similitude, with which I was mainly concerned. Itwould be well worthy of further discussion. The conclusion that I gave followson the basis of the usual Fourier equations for conduction of heat, in whichheat and temperature are regarded as sui generis. It would indeed be a paradoxif the further knowledge of the nature of heat afforded by molecular theory putus in a worse position than before in dealing with a particular problem. Thesolution would seem to be that the Fourier equations embody something as tothe nature of heat and temperature which is ignored in the alternative argumentof Dr. Riabouchinsky.

Although commentators have generally been hard on Rayleigh (Bridgman(1931: 11), for example, says that Rayleigh’s reply “is, I think, likely to leaveus cold”), Rayleigh is essentially correct. His dimensional argument (whetherused as prediction or explanation) takes as a premise that the target equationfor h is dimensionally homogeneous using L, T, �, and Q as fundamental


dimensions, just as in the case of the sphere falling through the viscousfluid, the explanans was that v stands in a relation to ρ s, ρ f , r, g, and η

that is dimensionally homogeneous using L, M, T, and F as fundamentaldimensions. This is indeed further information beyond the fact that thetarget equation for h is dimensionally homogeneous using L, M, and T (orequivalently L, T, and energy). This is not a case where more informationputs us in a worse position than before. Rather, more information puts us ina better position than before.

In the dimensional explanation of the derivative law governing the fallingsphere’s terminal velocity, the explanans includes that the law would stillhave held for any value of the constant of proportionality between forceand ma. Likewise, in Rayleigh’s dimensional explanation of the heat-flowequation, the explanans includes that the equation would still have held had(counterlegally) Boltzmann’s constant and the mechanical equivalent of heatdeparted from their actual values—or (counterlegally) had there been nosuch constants at all: had heat been a fluid (as caloric was thought to be).In the falling-sphere case, the rate of exchange between force and accelera-tion is explanatorily irrelevant because no force is producing an acceleration;the sphere has reached its terminal velocity. In the heat-flow case, the rateof exchange between heat and energy, or between temperature and randommolecular kinetic energy, is explanatorily irrelevant because no thermal en-ergy is being converted into mechanical energy (or any other sort of energy)or vice versa. For example, the fluid is not being called upon to do work(e.g., by pushing against a piston) and no internal molecular energy is beingtransformed into thermal energy. The thermal energy of the rigid body issimply contributing to the fluid’s thermal energy.28

In what I believe he intends to be an allusion to the Rayleigh-Riabouchinsky exchange, Bridgman (1931: 49–50) writes:

[H]ow [shall we] choose the list of physical quantities between which we are tosearch for a relation[?] We have seen that it does not do to merely ask ourselves“Does the result depend on this or that physical quantity?” for we have seenin one problem that although the result certainly does “depend” on the actionof the atomic forces, yet we do not have to consider the atomic forces in ouranalysis, and they do not enter the functional relation.

Here Bridgman appears to distinguish dimensional analysis from scientificexplanation (perhaps with scare quotes), whereas I regard dimensional anal-ysis as sometimes supplying explanations of derivative laws. In a derivationof the heat-flow equation from much more fundamental laws, the molecularnature of heat is invoked. However, the heat-flow equation (up to the unspec-ified function f(avc/κ)) can be accounted for by a dimensional explanation.The equation considers the phenomenon of heat flow purely at a phenomeno-logical level (e.g., in employing c and κ, which are properties of bulk matter),

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and (as Sedov 1959: 42 emphasizes) no thermal energy is converted into otherforms of energy or vice versa (since the fluid is ideal). Therefore, it is notevident that the equation actually depends on the molecular nature of heat.Indeed, part of the explanans in the dimensional explanation is precisely thatthe equation relating these various quantities to heat flow would still haveheld, even if (say) there had been no mechanical equivalent of heat.

8. Some Dimensional Explanations May Invoke Dimensional Meta-Laws

The explanans in a dimensional explanation is the fact that in order to forma quantity standing in a dimensionally homogeneous relation to a givenquantity, we need use nothing more than certain quantities (in the propercombination); we need no other quantities besides (some of ) those. Like theexplanans of a dimensional explanation, a symmetry principle also specifiescertain quantities to be dispensable to certain laws. For instance, the principleof symmetry under arbitrary spatial displacement specifies that the locationof any given event can be omitted from any law. Only the separation betweenevents matters to the law; the law privileges no particular spatial location.Space is “homogeneous.”

Such symmetry principles are frequently characterized as “meta-laws”—that is, as laws governing the first-order laws and thereby explaining whythey have certain features. For example, Wigner characterizes a symmetryprinciple as “a superprinciple which is in a similar relation to the laws ofnature as these are to the events” (1972: 10) and “as laws which the lawsof nature have to obey” (1985: 700; for similar remarks, see Morrison 1995and Feynman, 1967: 59, 83). In view of space-displacement symmetry, theinvariance of every first-order law under arbitrary spatial displacement is nota mere coincidence—a feature that every first-order law independently justhappens to possess. Rather, there is a common explanation of each first-orderlaw’s exhibiting this symmetry: each does so because all first-order laws haveto, as a matter of meta-law.

Indeed, symmetry meta-laws, as constraints on the first-order laws, arewidely taken as helping to account for various first-order laws, such as theconservation laws of momentum and energy.29 Since a symmetry meta-law“stands on its own, independent of any detailed theory of ” the particularkinds of forces there are (Weinberg 1992: 158), the first-order laws explainedby symmetry meta-laws hold independently of those details. For example,energy and momentum would still have been conserved even if (here comesa counterlegal) the particular forces had been different. As Wigner remarks,

[F]or those [conservation laws] which derive from the geometrical princi-ples of invariance it is clear that their validity transcends that of anyspecial theory—gravitational, electromagnetic, etc.—which are only loosely con-nected . . . (Wigner 1972: 13)


Thus, although energy and momentum conservation in a given system canbe deduced in classical physics from Newton’s laws of motion and the lawsgoverning the fundamental forces operating in that system, this deductiondoes not correctly explain why energy and momentum are conserved, sincethe details of the various particular force laws are explanatorily irrelevant.Energy’s conservation is not a coincidental byproduct of the sundry forcelaws.

As we have seen, dimensional explanations likewise reveal various deriva-tive laws to be independent of various details of the more fundamental lawsfrom which they can be deduced. This similarity suggests that perhaps somedimensional explanations proceed just like explanations from symmetry prin-ciples in that the explanans consists of meta-laws: principles transcendingthe first-order laws and imposing restrictions on the kinds of first-order lawsthere could have been.

To find a possible example, we must consider a derivative law that isplausibly explained entirely by meta-laws. Consider, for instance, the coordi-nate transformation laws, which have standardly been interpreted as purelykinematical—that is, as entirely independent of the particular kinds of forcesthere happen to be.30 Let us suppose that classical physics holds, and so thecoordinate transformation laws employ the Galilean transformations. Con-sider two inertial reference frames S and S′ (figure 6) where correspondingaxes are parallel and where the origin O′ in S′ moves relative to O (the originin S) with constant speed v in the x direction. Suppose that the momentwhen O coincides with O′ is the reference event marking time zero for bothreference frames. That is, their coinciding is an event with (x,y,z,t) coordi-nates (0,0,0,0) and with (x′,y′,z′,t′) coordinates (0,0,0,0). Then according toclassical physics31, the laws transforming any event’s unprimed into primedcoordinates are t′ = t, z′ = z, y′ = y, and x′ = x − vt.

I shall set out a dimensional explanation of why it is that in the x′ law, v

and t appear only to the first power and only together—that is, a dimensionalexplanation of why there is no term in which v figures without t, or vice versa,

Sy

v

O O′x, x′

z

y′

z′

S′

Figure 6

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or (for example) involving v2 or√

t. Our explanation involves both symmetrymeta-laws and dimensional meta-laws, and so shows that the explanandumtranscends any details of the first-order laws that are not required by meta-law.32

Let us start our explanation. If the transformation yields any event’s x′

as a function f of various quantities, then what quantities could they be?Take the dimensional meta-law to be that v and the event’s x, y, z, andt suffice to form a quantity that stands in a dimensionally homogeneousrelation f to the event’s x′.33 That transformations of the same form holdbetween all pairs of inertial frames follows from the “principle of relativity”:that the natural laws take the same form in all inertial frames (a meta-law).Furthermore, the “homogeneity of space” imposes a further demand on thefirst-order laws: that they take the same form in any two reference framesdiffering only in the location of their origin. Hence, the transformation fromS to S′ must also work for transforming from S′′ into S′, where S′′ differsfrom S only in its origin being displaced from O by some distance alongthe y axis. Hence, if y figures in the transformation from x to x′, then y′′

must likewise figure in the transformation from x′′ to x′. But O′ has thesame speed v relative to O as it does relative to O′′ and any event’s (x,y,z,t)will be the same as its (x′′,y′′,z′′,t′′) except for its y coordinate differing fromits y′′ coordinate. Therefore, if y figures in the transformation from x to x′,and if v, x, y, z, and t are the only quantities used by that transformationto yield x′, then the transformation from S will yield a different x′ valuefor a given event than the transformation from S′′ does. Therefore, y cannotfigure in the transformation from x to x′ (and, by similar reasoning, neithercan z).

Without elaboration, Einstein (1905/1952: 44) appeals to the homogeneityof space to show in addition that the transformation must be linear in x (thatis, no term in the transformation can raise x to higher than the first power).Here is one way that Einstein’s reasoning could be unpacked. Consider a rodplaced at time t along the x-axis so that in S, its left endpoint is at x1 and itsright endpoint is at x2. In S, its length is x2 − x1; in S′, its length is f(x2,v,t) −f(x1,v,t). Now instead suppose the same rod had been placed with its leftendpoint shifted to x1 + � in S. By the homogeneity of space, the rod wouldhave had the same length in S, so its right endpoint would have been atx2 + � in S. In S′, then, its endpoints would have been at f(x1 + �,v,t) andf(x2 + �,v,t). By the homogeneity of space, its length in S′ would have beenunaffected by the shift:

f (x2 + �, v, t) − f (x1 + �, v, t) = f (x2, v, t) − f (x1, v, t).

Hence

f (x2 + �, v, t) − f (x2, v, t) = f (x1 + �, v, t) − f (x1, v, t).


Dividing both sides by �, and then taking their limits as � → 0, we find

∂ f/∂x(x2) = ∂ f/∂x(x1).

That is, f ’s partial derivative with respect to x is the same whether evaluatedat x1 as at x2, for arbitrary x1 and x2. So ∂f /∂x is a constant. Therefore, fmust be linear in x.

Since x′ is to have the same dimensions as xα tβ vγ , our table is

x′ x t v

L 1 1 0 1

T 0 0 1 −1

From L: 1 = α + γ

From T: 0 = β − γ

Because the transformation must be linear in x, α can only be 0 or 1. If aterm in f (x,t,v) has α = 1, then β = γ = 0. If a term has α = 0, then β = γ =1. Therefore, any term in f (x,t,v) must be either proportional to x (withoutv, t) or proportional to vt (without x): the transformation law must be x′ =Ax + Bvt (for some dimensionless constants A, B).

Plausibly, this dimensional explanation uses a dimensional meta-lawalongside a symmetry meta-law to explain a feature of the transformationlaws. In view of being explained by these meta-laws, this feature transcendsthe particular force laws and other first-order laws there happen to be.

9. Conclusion

Dimensional explanations of derivative laws illustrate the fact that a deriva-tive law need not be explained by its deduction from more fundamental laws.Similar phenomena arise in connection with explanations in mathematics(see Lange forthcoming). Here is a very simple example. If we have threeobjects and each object is red or blue, then two of the objects are the samecolor. Why is that? We can deduce this fact by listing all 8 possibilities (RRR,RRB, RBR, RBB, BRR, BRB, BBR, BBB) and noting that in each case, atleast two of the objects are the same color. But this derivation fails to explainwhy the result obtains; for each possibility, there is the same reason why ithas two objects of the same color. Select one object. Either it is the samecolor as one of the others, or it is not. If it is not, then (since there are onlytwo possible colors) the other two objects must be the same color.

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This explanation shows that the explanandum (unlike the 8 possible com-binations) is not sensitive to various details of the case (such as the particularpair of colors involved). Indeed, the same sort of mathematical explanationcan be given of the fact that if we have four objects, and each of them is red,white, or blue, then two of the objects are the same color. (The analogousexplanation: Select an object. Either it is the same color as one of the others,or it is not. If not, then select another object. We know that it is not thesame color as the object first selected, and either it is the same color as oneof the remaining unselected objects, or it is not. If not, then since there areonly three colors, the two remaining objects must be the same color.)

Plainly, it is no coincidence that there is color duplication in both of thesecases: both involve more objects than colors.34 But the two separate, brute-force deductions of color duplication from the two lists of every possibilityin the two cases fails to remove the coincidence; that pair of deductions doesnot unify the two cases by tracing the color duplication in each case backto a feature that is also possessed by the other case. The two lists of everypossibility in two such cases fail to pick out the particular features of thetwo cases that are responsible for a given similarity or difference betweenthem. For instance, the two lists fail to explain why two of the objects mustbe the same color when there are three objects and two possible colors, butnot when there are three objects and three possible colors.

Even after a theorem in mathematics has been well-established, math-ematicians labor to discover new proofs of it partly because of the newexplanatory connections that those proofs may reveal. A similar phe-nomenon occurs in science. Dimensional explanations exemplify some ofthe kinds of explanatory contributions that can be made by different stylesof scientific reasoning.35

Notes1 Among the philosophical works on dimensional analysis are Bridgman (1931), Campbell

(1957), Causey (1967, 1969), Ellis (1966), Krantz, Luce, Suppes, and Tversky (1971), Luce(1971), and Laymon (1991). None characterizes dimensional analysis as capable of fundingscientific explanations. However, a reviewer kindly called my attention to Batterman’s work(e.g., Batterman 2002a, 2002b), where the explanatory contributions of dimensional analysis areexplored. Batterman understands dimensional analysis as the simplest species of “asymptoticexplanation”, and Batterman sees an asymptotic explanation as supplying a kind of under-standing that cannot be supplied, even in principle, by a derivation of the same explanandumfrom more fundamental laws. There are thus close similarities between Batterman’s themesand mine.

Batterman’s “asymptotic explanations” target relationships that emerge in the long run fora wide range of specific initial conditions (as the influence of those initial details dies out) andat the macroscopic level (where various microstructural details make negligible difference). LikeBatterman, I emphasize how dimensional explanations unify because they can ignore certainphysical details. However, although (as we shall see) dimensional arguments work by presup-posing that (at least to a sufficiently good approximation) a given phenomenon is independentof various parameters, I contend neither that an equation explained dimensionally must be the


result of some more fundamental theory taken to some limit nor that a dimensional argumentis explanatory by virtue of capturing the asymptote of some fundamental theory.

In addition, whereas Batterman emphasizes that the same asymptotic explanation appliesto (for instance) all simple pendulums whatever material their bobs are made of, I argue thatdimensional explanations allow connections to be drawn among much more disparate cases—for instance, among a pendulum, a body in free fall, and a water wave (see section 5). Thesecases have the same (or relevantly similar) dimensional architecture despite differing profoundlyin their physical features even when considered asymptotically—in the limit where the influencesof initial transients and microstructural details become negligibly small.

2 By a “derivative law”, I mean a logical consequence of natural laws alone (together withmathematical truths and other broadly logical necessities) that is not a fundamental law, butrather is explained by other laws. For example (according to classical physics), the centripetalforce law F = mω2r follows from and is explained by Newton’s second law of motion (togetherwith geometric facts). Some philosophers use the term “derivative law” to encompass somegeneralizations that do not follow from natural laws alone. For example, that all bodies fallingto Earth from a small height in conditions where all non-gravitational influences can be neglected(for example, with negligible air resistance) accelerate at approximately 9.8 m/s2 is sometimestermed a “derivative law” (“Galileo’s law of falling bodies”) even though it follows from morefundamental laws (such as Newton’s law of gravity and second law of motion) only when theyhave been supplemented by various non-laws, such as the accidental fact that Earth’s mass is5.98 × 1024 kg. I shall be concerned only with “derivative laws” that are naturally necessaryrather than accidental. A given derivative law’s scope may be rather narrow; the law may be aconsequence of more fundamental laws applied to a certain specific range of conditions.

3 In my forthcoming, I argue that when two derivative laws of nature are similar, despiteconcerning physically distinct processes, it may be that any correct scientific explanation of theirsimilarity proceeds by revealing their similarity to be no mathematical coincidence. Their simi-larity is explained mathematically—by mathematical similarities between the more fundamentalequations responsible for the two derivative laws.

4 That this derivation is generally regarded as explanatory is evident from common remarksalong these lines: “In Newton’s work the inverse square law appears as a means of accounting forthe observations of the solar system, particularly Kepler’s interpretation of Brahe’s observationsin terms of a relationship between periods and radii of orbits.” (Weinberg 1987: 6). But we willhave to consider whether the fact that this derivation explains why T2 = 4π2r3/GM suffices toensure that it explains why T ∝ r3/2.

5 Unfortunately, there are several common definitions of the term “dimensionally homoge-neous”. One is the definition that I just gave: that a relation is “dimensionally homogeneous”if and only if it holds in any system of units for the fundamental dimensions of the quantitiesso related. This definition may be found in articles and textbooks concerning a wide rangeof subjects—for example, coastal engineering (Hughes 1993: 27), engineering experimentation(Schenck 1979: 88), fluid mechanics (Shames 2002: 8), and pharmacokinetics (Rescigno 2003:23), as well as dimensional analysis (Langhaar 1951: 13 and 18; Laymon 1991: 148). (I willrefine this definition in section 7 so as to recognize that a given relation is dimensionally ho-mogeneous for a certain set of fundamental dimensions, such as {mass, length, and time}.) Asecond common definition is that a relation is “dimensionally homogeneous” if and only ifevery term in the equation has the same dimensions. Such definitions appear in Birkhoff 1950:80, Bridgman 1931: 41, and Furbish 1997: 116, for example. (However, Bridgman (1931: 37)also says that the assumption that a given relation holds for any system of units for the variousfundamental dimensions of the quantities so related “is absolutely essential to the treatment,and in fact dimensional analysis applies only to this type of equation.”) In section 6, I con-trast this second notion (which I term “dimensional consistency”) with the first (which I call“dimensional homogeneity”). As a third alternative, “dimensional homogeneity” is sometimesdefined directly in terms of the equation’s form, as described two paragraphs below in the maintext.

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6 This “motivation” fails to motivate the requirement that c > 0, which presumably arisesfrom the thought that two units cannot measure the same quantity if one increases while theother decreases (though they might then measure logically related quantities).

7 This criterion for units to specify the same quantity occasionally appears to depart fromordinary usage. For example, degrees Celsius and degrees Kelvin are ordinarily regarded asmeasuring the same quantity: temperature. But since their zeros do not coincide, a doubling ofdegrees Celsius—e.g., from 2◦C (275◦K) to 4◦C (277◦K)—does not coincide with a doublingof degrees Kelvin. However, if ◦K measures how far a temperature departs from absolute zero,then ◦C does not count as a unit for the same quantity as ◦K. (In laws such as the ideal-gas law(PV = nRT), T must be expressed in an absolute scale.) Nevertheless, on this criterion, ◦C and◦K are both units of temperature difference, since then the arbitrary zeros drop out.

8 Had T depended on the planet’s mass, then Kepler’s third law (that T ∝ r3/2 with thesame proportionality constant for every planet orbiting the sun) would not have held.

9 There might seem to be one difference: the salt’s being hexed explains nothing whereas theelements of the fundamental laws that do not contribute to explaining why T ∝ r3/2 holds helpto explain other facts (e.g., T ’s independence from m). However, the salt’s being hexed mightexplain how superstitious people treat the salt.

10 Here, I think, my view disagrees with Campbell’s. Although not discussing dimensionalarguments as explanations, he says that “for the application of the argument from dimensionseverything involved in the dynamical reasoning is required except the numerical values of no-dimensional magnitudes” (1957: 403, cf. 422).

11 That derivation, unlike the dimensional explanation, explains why the proportionalityconstant between T and

√(m/k) equals 2π .

12 Technically, this is the adiabatic bulk modulus (rather than the isothermal bulk modulus)because although the compressions and rarefactions are associated with temperature changes,they occur so rapidly that little heat can flow.

13 The L line in each case dictates that v is independent of λ—though for different reasonsin the two cases. However, we could not have reached this result had we also included the wave’samplitude in characterizing it. (But the M and T lines would have been unaffected.)

14 That two facts are naturally necessary does not suffice to make a given consequence ofthem no coincidence. For instance, nineteenth-century chemists believed it naturally necessarythat all noncyclic alkane hydrocarbons differ in molecular weight by multiples of 14 units, andthey also believed it naturally necessary that the atomic weight of nitrogen is 14 units. But theytermed it “coincidental” (albeit naturally necessary) that all noncyclic alkanes differ in molecularweight by multiples of the atomic weight of nitrogen. Noncyclic alkanes contain no nitrogen.See, for instance, van Spronsen 1969: 73–4 and Lange 2000: 203–7 as well as my (forthcoming)account of mathematical coincidences.

15 I have omitted a column for the angle from which the pendulum is released, since angleis usually taken to be a dimensionless quantity, and so dimensional analysis cannot impose anyconstraint on how it figures in the equation for the period. That is, dimensional analysis revealsonly that T is proportional to

√(l/g) times some unknown function of the initial angle.

16 Similarly, Hacking (1990) emphasizes that statistical reasoning identifies new targets ofexplanation (e.g., normal distributions).

17 The constant of proportionality turns out to be√

(2π ).18 Likewise Barenblatt (1987: 5): “The dimensions of both sides of any equation having

physical sense must be identical. Otherwise, the equation would no longer hold under a changeof fundamental units of measurement.” Douglas (1969: 3): “An equation about a real physicalsituation will be true only if all the terms are of the same kind and therefore have the samedimensions.”

19 Bridgman (1931: 42) and Birkhoff (1950: 83) give similar examples. Bridgman empha-sizes that “=” in the equation should be understood as numerical equality; obviously, if “=”required the same units on both sides, then trivially an equation would have to be dimensionallyconsistent.


Some might say that the example, though it follows logically from natural laws alone, isnot itself a law, and so fails to show that laws can be dimensionally inconsistent. In Lange 2000,I discuss the distinction between laws and naturally necessary non-laws.

20 Nor is dimensional consistency (plus truth) sufficient for dimensional homogeneity. Incgs electrostatic units, charge (like force) is not an independent dimension. It has dimensionL3/2 M1/2T−1. (One “electrostatic unit” is defined as the charge where the electrostatic forcebetween two point bodies so charged, 1 cm apart, equals 1 dyne.) Accordingly, Coulomb’s lawin these units (F = q1 q2 / r2) has no dimensional constant of proportionality. A change toother units would require the introduction of a dimensional proportionality constant. So thisexpression for Coulomb’s law is dimensionally consistent (in cgs electrostatic units) but notdimensionally homogeneous.

21 Shames (2002: 8), for example, invokes a “law of dimensional homogeneity” (that “ananalytically derived equation representing a physical phenomenon must be valid for all systemsof units”) as if it were one among the many contingent laws and meta-laws of nature.

22 In this, I think I agree with Campbell (1957: 366–9) and Ellis (1966: 117). That alllaws can be expressed in terms of dimensionally homogeneous relations seems akin to theview (advanced by Reichenbach, Hempel, Carnap, and others) that all fundamental laws canbe expressed without proper names or “local predicates” (i.e., predicates defined in terms ofparticular times, places, objects, events, etc.). Indeed, a similar intuition lies behind both ideas:the laws are too “general” to privilege any particular thing (whether an object or a unit). Formore on natural laws and nonlocal predicates, see Lange 2000.

23 Since it is trivial that every law can be expressed in terms of a dimensionally homoge-neous relation, I do not understand why those who believe in some non-trivial “principle ofdimensional homogeneity” qualify the principle; they say that we need an explanation “of why(most) numerical laws of physics are dimensionally invariant” (Krantz et al. 1971: 504, myitalics) or “of the prevalence of dimensionally invariant laws (Causey 1969: 256, my italics) or ofthe fact “that practically every law of physics is dimensionally invariant” (Causey 1967: 30, myitalics) or of the fact that “most, if not all, physical laws can be stated in terms of dimensionallyinvariant equations” (Luce 1971: 157).

24 Having included ρ s and an unspecified f(ρ f /ρ s) in the relation, there is no need also toinclude ρ f .

25 For another example of a law that has sometimes been regarded as having a staticalrather than a dynamical explanation (and so as independent of Newton’s second law), see myforthcoming2 concerning the law of the parallelogram of forces.

26 This yields the hydrodynamic answer when f(ρ f /ρ s) = (2/9) (1 – (ρ f /ρ s)).27 In Lange 2005, 2007 and forthcoming2, I have more extensively discussed the role of

counterfactuals (and other subjunctive facts) in explaining actuals. For particular attention tocounterlegals, see Lange 2009.

28 In contrast, take a case where thermal energy is converted to or from another form ofenergy. In giving a dimensional explanation for the gain in heat h of a body having mass mupon falling to the ground (without bouncing) from rest at height s, we must presume there tobe a dimensionally homogeneous relation between h and some subset of g, m, and s where heatis not given its own dimension. (No such dimensionally homogeneous relation is possible if heatis given its own dimension.) Dimensional reasoning then yields h ∝ mgs.

29 Symmetries are standardly taken as explaining conservation laws—see, for instance,Landau and Lifshitz 1976: 13; Wigner 1954: 199; Gross 1996: 14257; and Feinberg and Gold-haber 1963: 45. Not all philosophers agree that these arguments are explanatory (see, forinstance, Brown and Holland 2004: 1137–1138). I do not have the space here to examine themany interesting questions about whether these arguments are genuinely explanatory and, ifso, why. Lange 2007 and 2009 address these issues and specify the sense in which symmetryprinciples and the conservation laws they explain transcend the various particular force laws.

30 Brown (2005) is the most recent exponent of the heterodox tradition of regarding thecoordinate transformations as dynamical rather than as kinematical; Brown (2005) contains

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many further references to both traditions and an account of how the Galilean transformationscould be explained dynamically in classical physics—just as I shall now set out one way theycould be explained kinematically.

31 Of course, the relativistic transformation laws are the Lorentz transformations ratherthan the Galilean transformations. But for the purpose of showing how a dimensional explana-tion might proceed from meta-laws, it suffices to give an argument that could be explanatoryaccording to classical physics.

32 Some readers might regard the Galilean transformation laws as too obvious to requireexplanation. However, they are not obvious; in fact, they are false (see previous note). Soit is especially revealing to see where classical physics might say they come from, since onecomponent of that explanans must be false. This highlights the fact that neither symmetrymeta-laws nor dimensional meta-laws are knowable a priori. Neither consists of logical orconceptual necessities.

33 This turns out to be the premise that is violated in special relativity (see previous note); adimensionally homogeneous relation actually requires a further dimensional constant with thedimensions of speed.

34 In Lange forthcoming, I discuss the notion of a “mathematical coincidence” and itsrelation to mathematical and scientific explanation.

35 Thanks to Dina Eisinger, Martin Thomson-Jones, John Roberts, and Susan Sterrett forreading earlier drafts; to audiences at Kansas State University, the University of Maryland, andthe Triangle Philosophy of Science Ellipse; and to my daughter, Rebecca, for drawing some ofthe figures.

References

G.I. Barenblatt 1987. Dimensional Analysis. New York: Gordon and Breach.Robert W. Batterman 2002a. Asymptotics and the Role of Minimal Models. British Journal for

the Philosophy of Science 51: 21–38.Robert W. Batterman 2002b. The Devil in the Details. New York: Oxford University Press.Garrett Birkhoff 1950. Hydrodynamics. Princeton: Princeton University Press.P.W. Bridgman 1931. Dimensional Analysis. New Haven: Yale University Press.Harvey Brown 2005. Physical Relativity. Oxford: Clarendon.Harvey Brown and Peter Holland 2004. Dynamical versus Variational Symmetries: Understand-

ing Noether’s First Theorem. Molecular Physics 102: 1133–1139.Norman Robert Campbell 1957. Foundations of Science. New York: Dover.Robert L. Causey 1967. Derived Measurements and the Foundations of Dimensional Analysis,

Technical Report No. 5, Measurement Theory and Mathematical Models Reports.Eugene, Oregon: University of Oregon.

Robert L. Causey 1969. Derived Measurements, Dimensions, and Dimensional Analysis. Phi-losophy of Science 36: 252–70.

John F. Douglas 1969. An Introduction to Dimensional Analysis for Engineers. London: Pitman.Albert Einstein 1905/1952. On the Electrodynamics of Moving Bodies (1905), repr. in H.A.

Lorentz et al., The Principle of Relativity. New York: Dover, 1952, pp. 35–65.Brian Ellis 1966. Basic Concepts of Measurement. Cambridge: Cambridge University Press.Gerald Feinberg and Maurice Goldhaber 1963. The Conservation Laws of Physics. Scientific

American, 209 (October): 36–45.Richard Feynman 1967. The Character of Physical Law. Cambridge, MA: MIT Press.David J. Furbish 1997. Fluid Physics in Geology. New York: Oxford University Press.David Gross 1996. The Role of Symmetry in Fundamental Physics. Proceedings of the National

Academy of Sciences USA, 93: 14256–14259.Ian Hacking 1990. The Taming of Chance. Cambridge: Cambridge University Press.


Steven A. Hughes 1993. Physical Models and Laboratory Techniques in Coastal Engineeering:Advanced Series on Ocean Engineering, volume 7. Singapore: World Scientific.

Philip Kitcher 1993. The Advancement of Science. New York: Oxford University Press.David H. Krantz, R. Duncan Luce, Patrick Suppes, and Amos Tversky 1971. Foundations

of Measurement, Volume 1: Additive and Polynomial Representations. New York andLondon: Academic Press.

Henry E. Kyburg, Jr. 1965. Comment. Philosophy of Science 32: 147–51.L.D. Landau and E.M. Lifshitz 1976. Mechanics, 3rd ed. Oxford: Pergamon.Marc Lange 2000. Natural Laws in Scientific Practice. New York: Oxford University Press.Marc Lange 2005. How Can Instantaneous Velocity Fulfill Its Causal Role? Philosophical Review

114: 433–468.Marc Lange 2007. Laws and Meta-Laws of Nature: Conservation Laws and Symmetries. Studies

in History and Philosophy of Modern Physics 38: 457–81.Marc Lange 2009. Laws and Lawmakers. New York: Oxford University Press.Marc Lange forthcoming. What are Mathematical Coincidences (and Why Does It Matter)?

Mind.Marc Lange forthcoming2. A Tale of Two Vectors. Dialectica special issue on the metaphysics

of vectors.Henry Langhaar 1951. Dimensional Analysis and Theory of Models. New York: John Wiley &

Sons.Ronald Laymon 1991. Idealizations and the Reliability of Dimensional Analysis, in Paul T.

Durbin (ed.), Critical Perspectives on Nonacademic Science and Engineering. Bethlehem,PA: Lehigh University Press, pp. 146–80.

R. Duncan Luce 1959. On the Possible Psychophysical Laws. Psychological Review 66: 81–95.R. Duncan Luce 1971. Similar Systems and Dimensionally Invariant Laws. Philosophy of Science

38: 157–69.Margaret Morrison 1995. The New Aspect: Symmetries as Meta-Laws—Structural Metaphysics,

in Friedel Weinert (ed.), Laws of Nature: Essays on the Philosophical, Scientific, andHistorical Dimensions. Berlin: de Gruyter, pp. 157–88.

Lord Rayleigh 1915a. The Principle of Similitude. Nature 95: 66–68.Lord Rayleigh 1915b. The Principle of Similitude. Nature 95: 644.Aldo Rescigno 2003. Foundations of Pharmacokinetics. New York: Kluwer.Robert Resnick, David Halliday, and Kenneth Krane 1992. Physics, 4th ed. New York: Wiley.D. Riabouchinsky 1915. The Principle of Similitude. Nature 95: 591.Hilbert Schenck 1979. Theories of Engineering Experimentation, 3rd ed. New York: Hemisphere.L.I. Sedov 1959. Similarity and Dimensional Methods in Mechanics. London: Infosearch.Irving H. Shames 2002. Mechanics of Fluids, 4th ed. New York: McGraw-Hill.J.W. van Spronsen 1969. The Periodic System of Chemical Elements. Amsterdam: Elsevier.Stephen Weinberg 1987. Newtonianism and Today’s Physics, in S.W. Hawking and W. Israel

(eds.), Three Hundred Years of Gravitation. Cambridge: Cambridge University Press,5–16.

Stephen Weinberg 1992. Dreams of a Final Theory. New York: Pantheon.Eugene Wigner 1954. On Kinematic and Dynamic Laws of Symmetry. In Symposium on New

Research Techniques in Physics, July 15–29, 1952. Rio de Janeiro: Academia Brasileirade Ciencias, pp. 199–200.

Eugene Wigner 1972. Events, Laws of Nature, and Invariance Principles. In Nobel Lectures:Physics 1963–1970. Amsterdam: Elsevier, pp. 6–19.

Eugene Wigner 1985. Events, Laws of Nature, and Invariance Principles. In A. Zichichi (ed.),How Far Are We From the Gauge Forces – Proceedings of the 21st Course of theInternational School of Subnuclear Physics, Aug 3–14, 1983, Enrice Sicily. New Yorkand London: Plenum, pp. 699–708.

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